Do most Remington model 700's chambered for .308 require a 20 MOA rail for the rams? I will be using a Leupold FX-3, 25x40 scope if that helps. I hope that the base will make it easier to shoot the white buffalo as well. I did not like aiming 20-something feet above it.

No, unless the barrel isn't square with the receiver and points down. I'm only 11moa up from chickens to rams with my slowest ram loads out of my 260 Remington, and a 308 is typically only a couple more minutes at most. You can do the calculation on any ballistics calculator to see how many minutes there will be from your chicken to ram settings. Most target scopes have more than enough elevation. If you just want to have your scope in the middle of its adjustment range, you can get a rail with about 10 moa built into it.

The 20 MOA base would be more than you want for a lighter weight bullet. Over a 4" span of scope rings, that would be about .022 inches of height between the rings. That would be about 5 thickness of soda can if you are shimming. For my 7mm 162 grain, I used 3 pieces of soda can under the back ring and stayed well within the center adjustment of the scope. I know shimming is not a perfect solution but it worked for me. If you don't shim or use an angled base, the ram settings are going to be near the high end of scope adjustment where things can go just "wild".

I stuck the EGW 20MOA base on my do cheapo Remington 700 and I am glad to report that I was able to get a 200 yard zero. Now I need to drive up to the Whittington Center to see if it makes hitting the big buffalo any easier.
It is a good excuse to pop up there for a day or so.